Topics in optimization
Abstract
This thesis is a small contribution to the Mathematical framework of Multi-Objective Optimization. Specifically, we apply techniques from various branches of Mathematics and Statistics to develop a unique perspective on Multi-Objective Optimization. We first extend the deterministic theory of Multi-Objective Optimization to a probabilistic setting so as to incorporate uncertainty and randomness. We then generalize some of the fundamental notions such as Efficiency, Pareto-Optimality and Dominance to higher dimensions and expand the scope of some of the key results to the stochastic domain. Next, in the context of Multi-Objective Optimization, we propose a new approach of enlarging the efficient frontier via fixed point theory which helps to provide the decision maker with a whole new dimension of good choices which were previously unavailable. Further, we apply some of the results to problems in Financial Services and Banking focusing on the area of Credit Risk and Risk Appetite. In this regard, a new formulation of pricing of bad-debt using a combination of Multi-Objective Optimization, Time Series Analysis and Black-Scholes theory is proposed. The thesis also includes results on coordinate-free characterization, expansion and polynomial generation of the Pareto-Optimal solution space, polynomial parameterization of the efficient frontier and related topics.
Degree
Ph.D.
Advisors
Morin, Purdue University.
Subject Area
Industrial engineering|Operations research
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